Calculus Examples

$lim_{x \to \infty} x - \sqrt{x}$

Step 1

Multiply to rationalize the numerator.

$lim_{x \to \infty} \frac{(x - \sqrt{x}) (x + \sqrt{x})}{x + \sqrt{x}}$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Expand the numerator using the FOIL method.

$lim_{x \to \infty} \frac{x^{2} + x \sqrt{x} + x (- \sqrt{x}) - {\sqrt{x}}^{2}}{x + \sqrt{x}}$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to \infty} \frac{x^{2} - {(x^{\frac{1}{2}})}^{2}}{x + \sqrt{x}}$

Step 2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$lim_{x \to \infty} \frac{x^{2} - x^{\frac{1}{2} \cdot 2}}{x + \sqrt{x}}$

Step 2.2.3

Combine $\frac{1}{2}$ and $2$ .

$lim_{x \to \infty} \frac{x^{2} - x^{\frac{2}{2}}}{x + \sqrt{x}}$

Step 2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor.

$lim_{x \to \infty} \frac{x^{2} - x^{\frac{2}{2}}}{x + \sqrt{x}}$

Step 2.2.4.2

Rewrite the expression.

$lim_{x \to \infty} \frac{x^{2} - x^{1}}{x + \sqrt{x}}$

Step 2.2.5

Simplify.

$lim_{x \to \infty} \frac{x^{2} - x}{x + \sqrt{x}}$

Step 3

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x = \sqrt{x^{2}}$ .

$lim_{x \to \infty} \frac{\frac{x^{2}}{x} - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Cancel the common factor of $x^{2}$ and $x$ .

Tap for more steps...

Step 4.1.1.1

Factor $x$ out of $x^{2}$ .

$lim_{x \to \infty} \frac{\frac{x \cdot x}{x} - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.1.2

Cancel the common factors.

Tap for more steps...

Step 4.1.1.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{\frac{x \cdot x}{x^{1}} - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.1.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{\frac{x \cdot x}{x \cdot 1} - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.1.2.3

Cancel the common factor.

$lim_{x \to \infty} \frac{\frac{x \cdot x}{x \cdot 1} - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.1.2.4

Rewrite the expression.

$lim_{x \to \infty} \frac{\frac{x}{1} - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.1.2.5

Divide $x$ by $1$ .

$lim_{x \to \infty} \frac{x - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.1.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{x - \frac{x}{x}}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.2.2

Rewrite the expression.

$lim_{x \to \infty} \frac{x - 1 \cdot 1}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.1.3

Multiply $- 1$ by $1$ .

$lim_{x \to \infty} \frac{x - 1}{\frac{x}{x} + \sqrt{\frac{x}{x^{2}}}}$

Step 4.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $x$ .

$lim_{x \to \infty} \frac{x - 1}{1 + \sqrt{\frac{x}{x^{2}}}}$

Step 4.2.2

Cancel the common factor of $x$ and $x^{2}$ .

Tap for more steps...

Step 4.2.2.1

Raise $x$ to the power of $1$ .

$lim_{x \to \infty} \frac{x - 1}{1 + \sqrt{\frac{x^{1}}{x^{2}}}}$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$lim_{x \to \infty} \frac{x - 1}{1 + \sqrt{\frac{x \cdot 1}{x^{2}}}}$

Step 4.2.2.3

Cancel the common factors.

Tap for more steps...

Step 4.2.2.3.1

Factor $x$ out of $x^{2}$ .

$lim_{x \to \infty} \frac{x - 1}{1 + \sqrt{\frac{x \cdot 1}{x \cdot x}}}$

Step 4.2.2.3.2

Cancel the common factor.

$lim_{x \to \infty} \frac{x - 1}{1 + \sqrt{\frac{x \cdot 1}{x \cdot x}}}$

Step 4.2.2.3.3

Rewrite the expression.

$lim_{x \to \infty} \frac{x - 1}{1 + \sqrt{\frac{1}{x}}}$

Step 5

As $x$ approaches $\infty$ , the fraction $\frac{1}{x}$ approaches $0$ .

$lim_{x \to \infty} \frac{x - 1}{1 + \sqrt{0}}$

Step 6

Since its numerator is unbounded while its denominator approaches a constant number, the fraction $\frac{x - 1}{1 + \sqrt{0}}$ approaches infinity.

$\infty$